The Robinson–Ely Principle: A Constructive Proof of P = NP

This work presents the Robinson–Ely Principle as a constructive resolution framework for NP-complete constraint systems and develops a proof of P = NP. The construction represents a finite problem instance as a constraint graph whose vertices correspond to decision points and whose edges represent constraint relationships. For each unresolved decision point, an entropy-pressure quantity measures the proportion of admissible choices remaining relative to the available choice space. A corresponding resonance quantity determines the resolution ordering, prioritizing highly constrained decision points. The procedure then performs deterministic resolution, harmony alignment, and value propagation through the existing constraint structure. The resulting sequence of constraint states progressively reduces the unresolved state space without exhaustive enumeration of complete configurations. The proof establishes three properties of the construction: correctness, termination, and polynomial computational complexity. Correctness is established through preservation of the resolution invariant under admissible resolution and propagation. Termination follows from strict reduction of the unresolved decision set, yielding a bound of at most n successful resolution steps for n initially unresolved decision points. The computational analysis establishes polynomial bounds for graph construction, resonance ordering, propagation, and the complete sequence of resolution stages. These results provide a deterministic polynomial-time resolution procedure for the NP-complete constraint problem represented by the construction. By the defining property of NP-completeness, every language in NP can then be reduced to that problem in polynomial time, giving NP ⊆ P. Together with the standard inclusion P ⊆ NP, the construction yields P = NP. The paper presents the mathematical construction, its formal resolution procedure, correctness mechanism, termination argument, complexity analysis, and the resulting complexity-class implication. Sudoku, graph coloring, and the Traveling Salesman Problem are included as concrete instances through which the construction can be examined; they are not used as substitutes for the general complexity-class argument.